This project focuses on problems mainly in geometric analysis that can be formulated as partial differential equations of Monge-Ampere type, broadly understood. In general terms, the analytic and geometric techniques developed in this proposal should be useful to researchers working in geometry, physics, and elsewhere. On the one hand, deepening our understanding of canonical geometries on Kahler manifolds and Lagrangians in Calabi-Yau manifolds seems to be of interest to physicists trying to model the geometry of the universe. On the other hand, these canonical geometries have relations to a wide variety of established fields in mathematics. Moreover, Monge-Ampere type equations arise in a wide variety of problems in pure and applied mathematics and have a wide range of real-world applications, such as meteorology and optimal design of networks. Developing methods and techniques to construct and approximate solutions to such equations and to study their regularity could have applications in other instances where these equations appear. Finally, the Legendre transform is a classical tool in mathematics, mechanics, and economics, and seeking generalizations of this theory to other settings, as in this project, could find a broad range of applications.
A number of the equations proposed in this project are new, fall outside of the traditionally known Monge-Ampere type equations, and have exciting new geometric applications requiring new analytical tools. A novel feature in this research is to apply several different tools of microlocal analysis, traditionally pertaining to linear problems, to study these fully nonlinear equations. Another theme is to investigate novel relations between convex analysis and geometry and complex analysis and geometry. The problems investigated include: (1) Kahler-Einstein metrics with conic singularities. These metrics provide a new powerful analytic tool in algebraic and complex geometry. (2) The PI will study a new degenerate version of the special Lagrangian equation. It governs geodesics in the space of positive Lagrangians on a Calabi-Yau manifold. Understanding solutions of this equation requires new methods and will have applications to existence and uniqueness of special Lagrangians, singularities of Lagrangian mean curvature flow, the topology of the space of Lagrangians, and Lagrangian intersection theory. (3) The PI shall study the space of Kahler metrics using geodesics in the space, finite-dimensional Bergman approximations, Fourier integral operators with complex phase, and the metric space geometry of this space. (4) The PI is developing interactions between complex and convex geometry, including a differential theory for the polarity transform in parallel to the known theory for the Legendre transform. New equations of Hamilton-Jacobi and Monge-Ampere type that arise from this will be investigated as well as complex analogues. These equations provide new processes for interpolation between Banach spaces and new notions of optimal transportation.
